L(s) = 1 | + (1.08 − 0.903i)2-s − i·3-s + (0.366 − 1.96i)4-s + (0.660 + 2.13i)5-s + (−0.903 − 1.08i)6-s + (2.64 − 0.110i)7-s + (−1.37 − 2.46i)8-s − 9-s + (2.64 + 1.72i)10-s − 2.42i·11-s + (−1.96 − 0.366i)12-s + 5.02·13-s + (2.77 − 2.50i)14-s + (2.13 − 0.660i)15-s + (−3.73 − 1.43i)16-s − 6.95·17-s + ⋯ |
L(s) = 1 | + (0.769 − 0.639i)2-s − 0.577i·3-s + (0.183 − 0.983i)4-s + (0.295 + 0.955i)5-s + (−0.368 − 0.444i)6-s + (0.999 − 0.0416i)7-s + (−0.487 − 0.873i)8-s − 0.333·9-s + (0.837 + 0.546i)10-s − 0.732i·11-s + (−0.567 − 0.105i)12-s + 1.39·13-s + (0.741 − 0.670i)14-s + (0.551 − 0.170i)15-s + (−0.932 − 0.359i)16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75690 - 1.50030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75690 - 1.50030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.08 + 0.903i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.660 - 2.13i)T \) |
| 7 | \( 1 + (-2.64 + 0.110i)T \) |
good | 11 | \( 1 + 2.42iT - 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 0.903T + 31T^{2} \) |
| 37 | \( 1 - 8.02iT - 37T^{2} \) |
| 41 | \( 1 - 3.33iT - 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 + 3.31iT - 47T^{2} \) |
| 53 | \( 1 - 4.77iT - 53T^{2} \) |
| 59 | \( 1 + 8.38T + 59T^{2} \) |
| 61 | \( 1 + 4.38iT - 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.69iT - 79T^{2} \) |
| 83 | \( 1 + 9.72iT - 83T^{2} \) |
| 89 | \( 1 - 13.6iT - 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08318004004031884110342954199, −10.73323431820017239384242214019, −9.215855096117088982073414072080, −8.287883754064028177603196244051, −6.87764572170860223670951563242, −6.27041281761387631811314247181, −5.18952559641610808034186389641, −3.85700270584723454483522612678, −2.67880742823816582092501590926, −1.47644527439517556297086696982,
2.03616388233544810001828268543, 3.90025067564154138005881578489, 4.67411742662786220946041312751, 5.40821183958331901069724000988, 6.50870256062381891741376966576, 7.72916653724331101302649216136, 8.797498637609920928173384218908, 9.083995424527984312891227670415, 10.86483902325237428350898955233, 11.32354638931965164401026790326